366 
DR. J. C. FIELDS ON THE FOUNDATIONS OF THE 
identity is an expression linear in the constants which we shall represent by the 
notation „_j. The conditions imposed on the constants S by the identity ( 65 ) are 
then embodied in the identity 
i X = 0 .( 66 ) 
t = 1 q = l 
The n(^ —l) conditions imposed on the constants S by the equations 
= 0, q = l,2, 
1 ; t = i,2, 
n 
( 67 ) 
may or may not be independent of one another. 
The general rational function of {z, h) conditioned for the value 2 = a, by the orders 
of coincidence can, after the analogy of the function d>(z,'?() in formula ( 19 ), 
be represented in tlie form 
(g, u) 
{z-a,y- 
+ {{z-a,,u)) 
( 68 ) 
Furthermore, the first element in this expression can, after the analogy of formula • 
( 27 ), be represented in the form 
(g, U) 
(z 
2 * 
7 
,(0 
-g, n—t'^ 
~q,^^n t 
+ Z 
- 3 k 
'/ = 1 
(69) 
From ( 64 ) we see that the coefficients on the right-hand side of this identity 
are linear in terms of the arbitrary constants 8 }"^. 
Taking 'F {z, u) an integral rational function of {z, u) of arbitrarily assigned degree 
M in 2, and equating too the principal residue relative to the value z = <x> in the 
product 
n 3 k-^ 
^ 2 m), 
(70) 
J = ] q=l 
we see, on referring to formula ( 28 ) and the related text, that we thus obtain the 
necessary and sufficient conditions in order that F (2, u) may possess for the value 
2 = a set of orders of coincidence which are complementary adjoint to the orders of 
coincidence t/'‘\ the integerbeing assumed to have been taken sufficiently 
large. For the degree M of F (2, u) in 2 we shall find it convenient to choose a 
definite integer, and for this definite integer it will suit our present purpose to select 
the greatest degree in 2 of a rational function of (2, ii) which is compatible with the 
possession by the function of the orders of coincidence for the value 
2 = GO. We assume, then, that M has been so chosen, and at the same time we 
assume that the integers corresponding to the various values 2 = have all been 
taken sufficiently large. 
In § 4 we saw that the necessary and sufficient conditions in order that an integral 
rational function F (2, u) should, for the value 2 = co ^ have orders of coincidence 
